Education

Schmidt Number: S-5388

On-line since: 31st January, 2010

LECTURE IX

ARITHMETIC,
GEOMETRY,
HISTORY

14thAugust, 1923.

Arithmetic and
geometry, indeed all mathematics, occupy a unique position in education.
Education can only be filled with the necessary vitality and give
rise to a real interplay between the soul of the teacher and the soul
of the child, if the teacher fully realizes the consequences of his
actions and methods. He must know exactly what effect is made on the
child by the treatment he receives in school, or anywhere else.

Man is a being
of body, soul and spirit; his bodily nature is formed and moulded by the
spirit. The teacher, then, must always be aware of what is taking
place in the soul and spirit when any change occurs in the body, and
again, what effect is produced in the body when influences are
brought to bear on the life of spirit or soul.

Anything that
works upon the child's conceptual and imaginative faculties, anything that
is to say of the nature of painting or drawing which is then led over into
writing, or again, botany taught in the way indicated yesterday, all
this has a definite effect. And here, above all, we must consider a
higher member of man's being, a member to which I have already
referred as the etheric body, or body of formative forces. The human
being has, in the first place, his physical body. It is revealed to
ordinary physical sense-perception. Besides this physical body,
however, he has an inner organization, perceptible only to
Imaginative Cognition, a super-sensible, etheric body. Again he has an
organization perceptible only to Inspiration, the next stage of
super-sensible knowledge. (These expressions need not confuse us; they
are merely terms.) Inspiration gives insight into the so-called
astral body and into the real Ego, the Self of the human being.

From birth till
death, this etheric body, this body of formative forces which is the first
super-sensible member of man's being never separates from the physical
body. Only at death does this occur. During sleep, the etheric
organization remains with the physical body lying there in bed. When
man sleeps, the astral body and Ego-organization leave the physical
and etheric bodies and enter them again at the moment of waking.

Now it is the
physical and etheric bodies which are affected when the child is taught
arithmetic or geometry, or when we lead him on to writing from the basis
of drawing and painting. All this remains in the etheric body and its
vibrations persist during sleep. On the other hand, history and such
a study of the animal kingdom as I spoke of in yesterday's lecture
work only upon the astral body and Ego-organization. What results
from these studies passes out of the physical and etheric bodies into
the spiritual world during sleep.

If, therefore,
we are teaching the child plant-lore or writing, the effects are preserved
by the physical and etheric bodies during sleep, whereas the results
of history lessons or lessons on the nature of man are different, for
they are carried out into the spiritual world by the Ego and astral
body. This points to an essential difference between the effects
produced by the different lessons.

We must realize
that all impressions of an imaginative or pictorial nature made on the
child have the tendency to become more and more perfect during sleep.
On the other hand, everything we tell the child on the subject of
history or the being of man works on his organization of soul and
spirit and tends to be forgotten, to fade away and grow dim during
sleep. In teaching therefore, we have necessarily to consider whether
the subject-matter works upon the etheric and physical bodies or upon
the astral body and Ego-organization.

Thus on the one
hand, the study of the plant kingdom, the rudiments of writing and reading
of which I spoke yesterday affect the physical and etheric bodies. (I
shall speak about the teaching of history later on.) On the other
hand, all that is learnt of man's relation to the animal kingdom
affects the astral body and Ego-organization, those higher members
which pass out of the physical and etheric bodies during sleep. But
the remarkable thing is that arithmetic and geometry work upon both
the physical-etheric and the astral and Ego. As regards their role in
education arithmetic and geometry are really like a chameleon; by
their very nature they are allied to every part of man's being.
Whereas lessons on the plant and animal kingdoms should be given at a
definite age, arithmetic and geometry must be taught throughout the
whole period of childhood, though naturally in a form suited to the
changing characteristics of the different life-periods.

It is
all-important to remember that the body of formative forces, the etheric
body, begins to function independently when it is abandoned by the Ego
and astral body. By virtue of its own inherent forces, it has ever the
tendency to bring to perfection and develop what has been brought to it.
So far as our astral body and Ego are concerned, we are — stupid,
shall I say? For instead of perfecting what has been conveyed to
these members of our being, we make it less perfect. During sleep,
however, our body of formative forces continues to calculate,
continues all that it has received as arithmetic and the like. We
ourselves are then no longer within the physical and etheric bodies;
but supersensibly, they continue to calculate or to draw geometrical
figures and perfect them. If we are aware of this fact and plan our
teaching accordingly, great vitality can be generated in the being of
the child. We must, however, make it possible for the body of formative
forces to perfect and develop what it has previously received.

In geometry,
therefore, we must not take as our starting point the abstractions and
intellectual formulae that are usually considered the right groundwork.
We must begin with inner, not outer perception, by stimulating in the
child a strong sense of symmetry for instance.

Even in the case of the very youngest
children we can begin to do this. For example: we draw some figure on
the blackboard and indicate the beginning of the symmetrical line.
Then we try to make the child realize that the figure is not
complete; he himself must find out how to complete it. In this way we
awaken an inner, active urge in the child to complete something as
yet unfinished. This helps him to express an absolutely right
conception of something that is a reality. The teacher, of
course, must have inventive talent but that is always a very good
thing. Above all else the teacher must have mobile, inventive thought.

When he has
given these exercises for a certain time, he will proceed to others.
For instance, he may draw some such figure as this (left) on the
blackboard, and then he tries to awaken in the child an inner
conception of its spatial proportions. The outer line is then varied
and the child gradually learns to draw an inner form corresponding to
the outer (right). In the one the curves are absolutely
straightforward and simple. In the other, the lines curve outwards at
various points. Then we should explain to the child that for the sake
of inner symmetry

he must make in
the inner figure an inward curve at the place where the lines curve outwards
in the outer figure. In the first diagram a simple line corresponds to
another simple line, whereas in the second, an inward curve
corresponds to an outward curve.

Or again we
draw something of this kind, where the figures together form a harmonious
whole. We vary this by leaving the forms incomplete, so that the
lines flow away from each other to infinity. It is as if the lines
were running away and one would like to go with them. This leads to
the idea that they should be bent inwards to regulate and complete
the figure, and so on. I can only indicate the principle of the
thing. Briefly, by working in this way, we give the child an idea of
“a-symmetrical symmetries” and so prepare the body of
formative forces in his waking life that during sleep it elaborates
and perfects what has been absorbed during the day. Then the child
will wake in an etheric body, and a physical body also, inwardly and
organically vibrant. He will be full of life and vitality. This can,
of course, only be achieved when the teacher has some knowledge of
the working of the etheric body; if there is no such knowledge, all
efforts in this direction will be mechanical and superficial.

A true teacher
is not only concerned with the waking life but also with what takes place
during sleep. In this connection it is important to understand certain
things that happen to us all now and again. For instance, we may have
pondered over some problem in the evening without finding a solution.
In the morning we have solved the problem. Why? Because the etheric
body, the body of formative forces, has continued its independent
activity during the night.

In many respects
waking life is not a perfecting but a disturbing process. It is necessary
for us to leave our physical and etheric bodies to themselves for a
time and not limit them by the activity of the astral body and Ego.
This is proved by many things in life; for instance by the example
already given of someone who is puzzling over a problem in the
evening. When he wakes up in the morning he may feel slightly
restless but suddenly finds that the solution has come to him
unconsciously during the night. These things are not fables;
they actually happen and have been proved as conclusively as many
another experiment. What has happened in this particular case? The
work of the etheric body has continued through the night and the
human being has been asleep the whole time. You will say: “Yes,
but that is not a normal occurrence, one cannot work on such a
principle.” Be that as it may, it is possible to
assist the continued activity of the etheric body during sleep, if,
instead of beginning geometry with triangles and the like, where the
intellectual element is already in evidence, we begin by conveying a
concrete conception of space. In arithmetic, too, we must proceed in
the same way. I will speak of this next.

* * *

A pamphlet on
physics and mathematics written by Dr. von Baravalle (a teacher at the
Waldorf School) will give you an excellent idea of how to bring concreteness
into arithmetic and geometry. This whole mode of thought is extended
in the pamphlet to the realm of physics as well, though it deals
chiefly with higher mathematics. If we penetrate to its underlying
essence, it is a splendid guide for teaching mathematics in a way
that corresponds to the organic needs of the child's being. A
starting-point has indeed been found for a reform in the method of
teaching mathematics and physics from earliest childhood up to the
highest stages of instruction. And we can apply to the domain of
arithmetic what is said in this pamphlet about concrete conceptions
of space.

Now the point
is that everything conveyed in an external way to the child by arithmetic
or even by counting deadens something in the human organism. To start
from the single thing and add to it piece by piece is simply to
deaden the organism of man. But if we first awaken a conception of
the whole, starting from the whole and then proceeding to its parts,
the organism is vitalised. This must be borne in mind even when the
child is learning to count. As a rule we learn to count by being made
to observe purely external things — things of material,
physical life.

First we have
the 1 — we call this Unity. Then 2, 3, 4, and so forth, are added,
unit by unit, and we have no idea whatever why the one follows the other,
nor of what happens in the end. We are taught to count by being shown an
arbitrary juxtaposition of units. I am well aware that there are many
different methods of teaching children to count, but very little
attention is paid nowadays to the principle of starting from the
whole and then proceeding to the parts. Unity it is which first of
all must be grasped as the whole and by the child as well. Anything
whatever can be this Unity. Here we are obliged to illustrate
it in a drawing. We must therefore draw a line; but we could use an
apple just as well to show what I shall now show with a line.

This then is 1.
And now we go on from the whole to the parts, or members. Here then we have
made of the 1 a 2, but the 1 still remains. The unit has been divided into
two. Thus we arrive at the 2. And now we go on. By a further
partition the 3 comes into being, but the unit always remains as the
all-embracing whole. Then we go on through the 4, 5, and so on.
Moreover, at the same time and by other means we can give an idea of
the extent to which it is possible to hold together in
the mind the things that relate
to number and we shall discover how really limited man is in his
power of mental presentation where number is concerned.

In certain
nations to-day the concept of number that is clearly held in the mind's
eye only goes up to 10. Here in this country money is reckoned up to 12.
But that really represents the maximum of what is mentally visualised
for in reality we then begin over again and in fact count what has been
counted. We first count up to 10, then we begin counting the tens, 2
times 10=20, 3 times 10=30. Here we are no longer considering
the things themselves. We begin to calculate by using number itself,
whereas the more elementary concept requires the things themselves to
be clearly present in the mind.

We are very proud of the
fact that we are far advanced in our methods of counting compared
with primitive peoples who depend on their ten fingers. But there is
little foundation for this pride. We count up to 10 because we sense
our hands as members. We feel our two hands symmetrically with their
10 fingers. This feeling also arises and is inwardly experienced by
the child, and we must call forth the sense of number by a transition
from the whole to the parts. Then we shall easily find the other
transition which leads us to the counting in which one is added to
another. Eventually, of course, we can pass on to the ordinary 1, 2,
3, etc. But this mere adding of one or more units must only be
introduced as a second stage, for it has significance only here in
physical space, whereas to divide a unity into its members has an
inner significance such that it can continue to vibrate in the
etheric body even though quite beneath our consciousness. It is
important to know these things.

Having taught
the child to count in this way, the following will also be important. We
must not pass on to addition in a lifeless, mechanical way merely adding
one item to another in series. Life comes into the thing when we take our
start not from the parts of the addition sum but from the sum total
itself. We take a number of objects; for example, a number of little
balls. We have now got far enough in counting to be able to say: Here
are 14 balls. Now we divide them, extending this concept of a part
still further. Here we have 5, here 4, here 5 again. Thus we have
separated the sum into 5 and 4 and 5. That is, we go from the sum to
the items composing it, from the whole to the parts. The method we
should use with the child is first to set down the sum before him and
then let the child himself perceive how the given sum can be divided
into several items.

This is
exceedingly important. Just as to drive a horse we do not harness him
tail foremost, so in the teaching of arithmetic we must have the right
direction. We must start from a whole which is always actually
present, from a reality, from what is present as a whole and then
pass on to the separate parts; later, we find our way to the ordinary
addition sum.

Continuing thus,
from the living whole to the separate parts, one touches the reality
underlying all arithmetical calculations: i.e., the
setting in vibration of the body of formative forces. This body needs
a living stimulus for its formative activity and once energised it
will continually perfect the vibrations without the need of
drawing upon the astral body and Ego-organization with their
disturbing elements.

Your teaching
work will also be essentially enhanced and vivified if you similarly
reverse the other simple forms of calculation. To-day, one might say,
they are standing on their heads and must be reversed. Try, for
instance, to bring the child to say: “If I have 7, how much
must I take away to get 3,” instead of “What remains over
if I take 4 from 7?” That
we have 7 is the real thing and that 3 remains is also real; how much
must we take away from 7 to get 3? Beginning with this form of
thought we stand in the midst of life, whereas with the opposite form
we are dealing with abstractions. Proceeding in this way, we can
easily find our way further. Thus, once more, in multiplication
and division we should not ask what will result when we divide 10
into two parts, but how must we divide 10 to get the number 5. The
real aspect is given; moreover in life we want eventually to get at
something which has real significance. Here are two children, 10
apples are to be divided among them. Each of them is to get 5. These
are the realities. What we have to deal with is the abstract part
that comes in the middle. Done in this way, things are always
immediately adapted to life and should we succeed in this, the result
will be that what is the usual, purely external way of adding, by
counting up one thing after another with a deadening effect upon the
arithmetic lessons, will become a vivifying force, of especial
importance in this branch of our educational work. And it is evident
that precisely by this method we take into account the sub-conscious
in man, that is, the part which works on during sleep and which also
works subconsciously during the waking hours. For one is aware of a
small part only of the soul's experience; nevertheless the rest is
continually active. Let us make it possible for the physical and
etheric bodies of the child to work in a healthy way, realizing that
we can only do so if we bring an intense life, an awakened interest
and attention, especially into our teaching of arithmetic and geometry.

The question
has arisen during this Conference as to whether it is really a good thing
to continue the different lessons for certain periods of time as we do
in the Waldorf School. Now a right division of the lessons into
periods is fruitful in the very highest degree. “Period”
teaching means that one lesson shall not perpetually encroach upon
another. Instead of having timetables setting forth definite hours:
— 8 – 9, arithmetic, 9 – 10, history, religion, or
whatever it may be, we give one main lesson on the same subject for
two hours every morning for a period of three, four, or five weeks.
Then for perhaps five or six weeks we pass on to another subject, but
one which in my view should develop out of the other, and which is
always the same during the two hours. The child thus concentrates
upon a definite subject for some weeks.

The question
was asked whether too much would not be forgotten, whether in this way
the children would not lose what they had been taught. If the lessons
have been rightly given, however, the previous subject will go on
working in the subconscious regions while another is being
taken. In “period” lessons we must always reckon with the
subconscious processes in the child. There is nothing more fruitful
than to allow the results of the teaching given during a period of
three or four weeks to rest within the soul and so work on in the
human being without interference.

It will soon
be apparent that when a subject has been rightly taught and the time comes
round for taking it up again for a further period it emerges in a different
form from what it does when it has not been well taught. To make the
objection that because the subjects will be forgotten it cannot be
right to teach in this way, is to ignore the factors that are at
work. We must naturally reckon on being able to forget, for just
think of all we should have to carry about in our heads if we could
not forget and then remember again! The part played by the fact of
forgetting therefore as well as the actual instruction must be
reckoned with in true education.

This does not
mean that it should be a matter for rejoicing whenever children forget.
That may safely be left to them! Everything depends on what has so passed
down into the subconscious regions, that it can be duly recalled.
The unconscious belongs to the being of man as well as the conscious.
In regard to all these matters we must realize that it is the task of
education to appeal not only to the whole human being, but also to
his different parts and members. Here again it is essential to start
from the whole; there must first be comprehension of the whole and
then of the parts. But to this end it is also necessary to take one's
start from the whole. First we must grasp the whole and then the
parts. If in counting we simply place one thing beside another, and
add, and add, and add, we are leaving out the human being as a whole.
But we do appeal to the whole human being when we lay hold of Unity
and go from that to Numbers, when we lay hold of the sum, the
minuend, the product and thence pass on to the parts.

* * *

The teaching
of history is very open to the danger of our losing sight of the human
being. We have seen that in really fruitful education everything must be
given its right place. The plants must be studied in their connection with
the earth and the different animal species in their connection with
man. Whatever the subject-matter, the concrete human element must be
retained; everything must be related in some way to man.

But when we
begin to teach the child history, we must understand that at the age when
it is quite possible for him to realize the connection of plant-life with
the earth and the earth itself as an organism, when he can see in the
human being a living synthesis of the whole animal kingdom, he is
still unable to form any idea of so-called causal connections in
history. We may teach history very skilfully in the ordinary sense,
describing one epoch after another and showing how the first is the
cause of the second; we may describe how in the history of art,
Michelangelo followed Leonardo da Vinci, for instance, in a natural
sequence of cause and effect. But before the age of twelve, the child
has no understanding for the working of cause and effect, a principle
which has become conventional in more advanced studies. To deduce the
later from the earlier seems to him like so much unmusical strumming
on a piano, and it is only by dint of coercion that he will take it
in at all. It has the same effect on his soul as a piece of stone
that is swallowed and passes into the stomach. Just as we would never
dream of giving the stomach a stone instead of bread, so we must make
sure that we nourish the soul not with stones but with food that it
can assimilate. And so history too, must be brought into connection
with Man and to that end our first care must be to awaken a
conception of the historical sequence of time in
connection with the human being.

Let us take
three history books, the first dealing with antiquity, the second with
the Middle Ages, and the third with our modern age. As a rule, little
attention is paid to the conception of time in itself. But suppose I
begin by saying to the child: “You are now ten years old, so
you were alive in the year 1913. Your father is much older than you
and he was alive in the year 1890; his father, again, was alive in
1850. Now imagine that you are standing here and stretching your arm
back to someone who represents your father; he stretches his arm back
to his father (your grandfather), now you have reached the year
1850.” The child then begins to realize that approximately one
century is represented by three or four generations. The line of
generations running backwards from the twentieth century brings him
finally to his very early ancestors. Thus the sixtieth generation
back leads into the epoch of the birth of Christ. In a large room it
will be possible to arrange some sixty children standing in a line,
stretching an arm backwards to each other. Space is, as it were,
changed into time.

If the teacher
has a fertile, inventive mind, he can find other ways and means of
expressing the same thing — I am merely indicating a principle.
In this way the child begins to realize that he himself is
part of history; figures like Alfred the Great, Cromwell and others
are made to appear as if they themselves were ancestors. The whole of
history thus becomes an actual part of life at school when it is
presented to the child in the form of a living conception of time.

History must
never be separated from the human being. The child must not think of it
as so much book-lore. Many people seem to think that history is something
contained in books, although of course it is not always quite as bad
as that. At all events, we must try by every possible means to awaken
a realization that history is a living process and that man himself
stands within its stream.

When a true
conception of time has been awakened, we can begin to imbue history with
inner life and soul, just as we did in the case of arithmetic and geometry,
by unfolding not a dead but a living perception. There is a great deal
of quibbling to-day about the nature of perception, but the whole
point is that we must unfold living and not dead perception. In the
symmetry-exercises of which I spoke, the soul actually lives in the
act of perception. That is living perception.
Just as our aim is to awaken a living perception of space, so must
all healthy teaching of history given to a child between the ages of
nine and twelve be filled with an element proceeding in this case not
from the qualities of space, but from the qualities of heart and soul.

The history
lessons must be permeated through and through with a quality proceeding
from the heart. And so we must present it as far as possible in the form
of pictures. Figures, real forms must stand there and they must never be
described in a cold, prosaic way. Without falling into the error of
using them as examples for moral or religious admonition, our
descriptions must nevertheless be coloured with both morality and
religion. History must above all lay hold of the child's life of
feeling and will. He must be able to enter into a personal
relationship with historic figures and with the modes of life
prevailing in the various historical epochs. Nor need we confine
ourselves merely to descriptions of human beings. We may, for
instance, describe the life of some town in the twelfth century, but
everything we say must enter the domains of feeling and will in
the child. He must himself be able to live in the events, to form his
own sympathies and antipathies. His life of feeling and will must be
stimulated.

This will show
you that the element of art must everywhere enter into the teaching of
history. The element of art comes into play when, as I often describe it,
a true economy is exercised in teaching. This economy can be exercised
if the teacher has thoroughly mastered his subject-matter before he
goes into the classroom; if it is no longer necessary for him to
ponder over anything because if rightly prepared it is there
plastically before his soul. He must be so well prepared that the
only thing still to be done is the artistic moulding of his lesson.
The problem of teaching is thus not merely a question of the pupil's
interest and diligence, but first and foremost of the teacher's
interest, diligence and sincerity.

No lesson
should be given that has not previously been a matter of deep experience
on the part of the teacher. Obviously, therefore, the organization of the
body of teachers must be such that every teacher is given ample time to
make himself completely master of the lessons he has to give.

It is a
dreadful thing to see a teacher walking round the desks with a book in
his hands, still wrestling with the subject-matter. Those who do not
realize how contrary such a thing is to all true principles of education
do not know what is going on unconsciously in the souls of the children,
nor do they realize the terrible effect of this unconscious experience.
If we give history lessons in school from note-books, the child comes
to a certain definite conclusion, not consciously, but unconsciously.
It is an unconscious, intellectual conclusion, but it is deeply
rooted in his organism: “Why should I learn all these things?
The teacher himself doesn't know them, for he has to read from notes.
I can do that too, later on, so there is no need for me to learn them
first.” The child does not of course come to this conclusion
consciously, but as a matter of fact when judgments are rooted
in the unconscious life of heart and mind, they have all the greater
force. The lessons must pulsate with inner vitality and freshness
proceeding from the teacher's own being. When he is describing
historical figures for instance the teacher should not first of all
have to verify dates. I have already spoken of the way in which we
should convey a conception of time by a picture of successive
generations. Another element too must pervade the teaching of
history. It must flow forth from the teacher himself. Nothing must be
abstract; the teacher himself as a human being must be the vital factor.

It has been
said many times that education should work upon the being of man as a whole
and not merely on one part of his nature. Important as it is to consider
what the child ought to learn and whether we are primarily concerned with
his intellect or his will, the question of the teacher's influence is
equally important. Since it is a matter of educating the whole nature
and being of man, the teacher must himself be “man” in
the full sense of the word, that is to say, not one who teaches and
works on the basis of mechanical memory or mechanical knowledge, but
who teaches out of his own being, his full manhood. That is the
essential thing.